Atlas › Modular forms › Weight 2 › Level 806 › 806.2.a

Newspace 806.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(806))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
806.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + O(q^{8})$	+ + +	$-1$	806b1 rank 1
806.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$	806a1 rank 1
806.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} + q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	− − +	$-1$	806d1 rank 1
806.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$	806f1 rank 0
806.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} - 3q^{7} + O(q^{8})$	− − +	$-1$	806c1 rank 1
806.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 3q^{5} + q^{6} - q^{7} + O(q^{8})$	− − −	$+1$	806e1 rank 0
806.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	+ − −	$-1$
806.2.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} - 4q^{7} + O(q^{8})$	− + −	$-1$
806.2.a.i	3	degree 3	$3q + 3q^{2} + 3q^{3} + 3q^{4} - 5q^{5} + 3q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$
806.2.a.j	5	degree 5	$5q - 5q^{2} - 3q^{3} + 5q^{4} + 5q^{5} + 3q^{6} + 8q^{7} + O(q^{8})$	+ − +	$+1$
806.2.a.k	5	degree 5	$5q - 5q^{2} - q^{3} + 5q^{4} + 3q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ + −	$+1$
806.2.a.l	6	degree 6	$6q + 6q^{2} + 5q^{3} + 6q^{4} + 3q^{5} + 5q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
806.2.a.a	$x + 39$
806.2.a.b	$x + 73$
806.2.a.c	$x + 151$
806.2.a.d	$x - 71$
806.2.a.e	$x + 19$
806.2.a.f	$x - 25$
806.2.a.g	$x^{2} + 28x - 4009$
806.2.a.h	$x^{2} + 134x + 2489$
806.2.a.i	$x^{3} + 51x^{2} - 330x - 23563$
806.2.a.j	$x^{5} - 111x^{4} - 2691x^{3} + 382692x^{2} + 3147988x - 32756179$
806.2.a.k	$x^{5} - 109x^{4} - 5435x^{3} + 43012x^{2} + 2248156x + 2646683$
806.2.a.l	$x^{6} - 177x^{5} - 3481x^{4} + 524820x^{3} + 11285106x^{2} - 110996219x - 1189662206$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(806))$	29
Old forms	80
Cusp forms $S_{2}(\Gamma_0(806))$	109
Eisenstein series $E_{2}(\Gamma_0(806))$	7
Modular forms $M_{2}(\Gamma_0(806))$	116

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(806)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(403))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.541 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 2, 13, 26, 31, 62, 403, 806]: [0, 1, 1, 5, 3, 10, 38, 116] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 8 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(806, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 806:2 1000

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